Weak mixing in rational billiards

TL;DR

This paper characterizes rational polygons with weakly mixing billiard flows in almost every direction, confirming a longstanding conjecture by Gutkin, and links this to properties of translation surfaces.

Contribution

It provides a complete characterization of rational polygons and translation surfaces that exhibit weak mixing in almost every direction, resolving Gutkin's conjecture.

Findings

Rational polygons not almost integrable are weakly mixing in almost every direction.

Translation surfaces without an affine circle factor are weakly mixing in almost every direction.

The results connect billiard dynamics with geometric properties of translation surfaces.

Abstract

We completely characterize rational polygons whose billiard flow is weakly mixing in almost every direction as those which are not almost integrable, in the terminology of Gutkin, modulo some low complexity exceptions. This proves a longstanding conjecture of Gutkin. This result is derived from a complete characterization of translation surfaces that are weakly mixing in almost every direction: they are those that do not admit an affine factor map to the circle.